in the RHS of (1) must be equal to some in the LHS of (1). this means has to be a normal subgroup of

conversely, if is a normal subgroup of then and hence (1) holds.

That's much easier than I was anticipating. I do have one further question - why is RG a Free Ring? According to Wiki, Group Rings are Free Modules, things I know next to nothing about, but Free Rings are not mentioned...